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Full text of "Treatise On Analysis Vol-Ii"

176 XIII INTEGRATION (f) Suppose that X is compact and that the measure ft is diffuse (13.18). Show that, for every neighborhood V of 0 in <?"(X, ft) and every function / e ^(X, ft), there exists an integer n such that all the functions af (where a is any real number) belong to y _j_ . . . _|_ v (n summands). Deduce that every continuous linear form on «^(X, ft) is identically zero, and hence that every vector subspace of finite codimension in , ft) is dense in <^(X, ft). 3. Suppose that X is compact and the measure ft diffuse (1 3.18). (a) Let (fynzi be a Hilbert basis of L^(X, ft). Show that, for each real number 8 > 0, there exists a compact subset Y of X such that ft(X — Y) ^ 8 and the sequence (/n)n£2 is total in LR(¥, ftY). (By using Problems 2(e) and 2(f), show that there exists a sequence of linear combinations of the /„ (n ^> 2) which converges in measure to /i ; then use Problem 2(c) and EgorofF s theorem.) (b) Show that there exists a bounded measurable function h such that the sequence (%fn)nZ2 is total in Lj(X, ft). (Choose h > 0 such that/i/A $ JS?J(X, ft), and then show that no nonnegligible function can be orthogonal to hfn for all n ^ 2.) 4. Let;? be a finite real number *>l. A subset H of ^?R(X, ft) is said to be equi-integrable if, for each e> 0, there exists a compact subset K of X such that \f\p dp, rge for all /e H, and a real number 8 > 0 such that f \f\p dp-^s for all /e H and all integrable sets A of measure ft(A) <£ S. (a) On an equi-integrable set H, show that the topology of convergence in measure is the same as that defined by the seminorm Np . Is the conclusion true if H is merely bounded in -S?p? (b) A sequence (/„) in -S?fc(X, ft) is convergent if and only if it is equi-integrable and convergent in measure. (c) Suppose that the measure ft is bounded and that ft(X) = 1 . Let (/„) be a sequence of functions belonging to J2P£, and suppose that lim \fndfji= lim \\&fn\ dft = l, n-*ooj n-»ooj lim Nid - i/n|) = 0. Show that lim f | Sfn\ dp. = 0. (Use (b).) n-+oo n-fooj (d) With ft as in (c), let (/„) be a sequence of functions belonging to & c suc^ tnat lim f/n</ft=lim f|/n|^« lim f|/;|1/2 dp,= 1. n-*oo J n-+oo J n-*ooj Show that HmNi(l— /n) = 0. (Reduce to the situation of (c) above by using the n-+oo Cauchy-Schwarz inequality; write H - fn\^\l- \fn\\ + \\fn\-fn\ and 5. Let F be a real-valued function with period 1 on R which is integrable on the unit interval I = [0, 1 ] (with respect to Lebesgue measure). (a) If /is any bounded measurable function on [0, 1], then lim f /(/)FOi/) dt = ( f F (0 dt]( f' «-»»Jo \Jo / woection 13.8,